Wave filter



R. P. LERCY WAVE FILTER April 8, 1952 2 v SHEETS-SHEET 1 Filed Feb. 5, 1947 Fig-(PRIUR ART) Fig a (5mm: ART) mm am I Z! L [/v w-wro/a P056??? HERE: 45207 3 4/7'dE VEYS April 8, 1952 R p LEROY I 2,591,838

WAVE FILTER Filed Feb. 5, 1947 2 SHEETS-$HEET 2 x Tl-i i I Hg 1" i i as ,5 [NVfN/ZW? i; 17. 065?! Esme: Asia) Patented Apr. 8, 1952 $591,838 AVE FILTER Robert Pierre Leroy, Paris, France, assignor to Compagnie Generale dElectricite, Paris;

France,;a corporation of France Application February 5, 1947, Serial No. 726,503

In France August .1945.

7 Section 1', Public Law 690, August '8, i946 Patent expires August 23', 1965.

This preseiitinvention elates to electric filters used in the application of weak currents as in communication circuits.

'I'hepresent invention has for its objectto: provide symmetrical units of the latticeor bridge F type an arrangement which, as is known, makes it-possible to produce symmetrical filters of. the: most general kind, obtained by means of a small-i:

er'number of elements than with 'equivalentun'its df the' type'of filters heretoforeusually employed having two sets of twoidentical reactances.

-The units whichform-thesubject'matter of the present 'i'nvention are characterised by the fact that -they coniprise two sets of reactances ar.-.. ranged in lattice form, one of said sets consisting of two. identical oppositely positioned reactances,i and the other set consisting of two unequallre-l actances, while the values of the constituentelements of each of these four reactances are .de-. termined in the manner explained below, so as to obtain in their construction a saving of elements of atleast one circuit element ascompared with the equivalent unit of the mentioned type as heretofore usually employed.

Although the mode of deltermination of these filters may be fixed independently of that of. the

type of filters'heretoforeusually employed having:

two setsof two identical reactances it is preferable,.forthe.sake of clarity, to refer to the latter, the prepernes'or wmeh are well known, and to deduce-the constitution of the new filters her'e'v described froin'that of the old'filters ofithe prior art, to which they are 'e1e'e'rr1ee11y equivalentl .UsuaLfilters. in ..lattice form;- as heretofore. em

ployed comprise two. inputlterminals .A, .Itwoqut putl ter inals "'0; lqj'a'nd {fou .re'actances', in. 'airszl; 'ziiafid ze," zz. gure 1' represents an e a ple o .t e. :ebo. e.:ar an ni Moreover; such-filters as heretofore'used are symmetricakthat is.to. .say that their operation slnot '.chan g 'ed if. inin'als and A," B as output terminals.

C, D "are taken as input ter- 7 the new. unit of the" present invention shbw'n" inflilgure'z one of the sets of; oppositely positionedreactances, zzjforjexample of Fig. '1, has

been-retainedywhilst the other is'etjof opposed reactance's is replaced. by two unequal 'reactances ia d. a

see that thefcondition of electric equivalence be:

Tl-ie interchangeofreactances X1 and X2 does not: alter the intensity of the output currentnorthe voltage difference at the output terminals; but only, the values of the voltagedifierences. V's- VA and. Vn-VB. The position of the re-, actances X1 and X2 in their own part of the unit is therefore; unimportant, and their interchange; is possible'in all circuit diagrams which willbe; shown. The different expressions obtained for. v the reactances of the partof the unit they. constitute, may lee-individually attributed to them;

arbitrarily. With a given lattice structure in which Z1 and Zz are known the relation (,1) makes it possible to deduce therefrom a large number-of equivalent.

structures .for reactances X1 and X2, inasmuch as a certain additional freedom is'available.

However, if Z1 and Z2 are known functions of. the'angular frequency of the signals to be transmitted, it is nevertheless necessary that the func-.

tions X1 and X2 of w which were first determined from the relation (1) should represent these reactan'ces, and this leads to certain restrictions in. regard .to their constitution" For instanceit'is notp'ossible totake arbitrarily for X1 any func-' tionof e, because the value of X2 which would be. deduced therefrom could. not generally be. rep-' resented by a practically realisable electrical'sys tem. Nevertheless, a certain amount of free-- dom is "left, which m'aybeturned to account to simplify'thest'ructure of the filter. 1

The manner in which the elements of lpe'rsY. according to the present invention, are deter mined will therefore be eXplained starting from the mentioned usual .type oi -filter units with two distinct sets of impedance Z1, Z2, whether:

such filters consist of coils and condensers, or whether they. comprise eleetromechanical elements such as piezo electric crystals.

We shall, here consider band filters. The" present -invention,.however, is not limited to'the' above cases and it may beapplied to all'filters properly so-called, except all-band filters.

We shall confine ourselves to filters with a singlepassband' It will be supposed, therefore,

that zeroand infinite frequencies are .in the sup'-' pressed bands T whichf are eliminated, and that the reactances Z1 and Z2 of thetwo sets of branches have. sirnultaneously either 'z'eros or.

poles for .such frequencies.

polein. the suppressedibands.

1 .Within the limits of these assumptions, four cases have to be considered, according to thle "Finally," we shall confine ourselves. a t e first? place. to the case where thereis nootherfzeroor and values of Z1 and Z2 at the frequencies and Case I.-Z1 and Z2 have zero values for the.

good if X1 and X2 are simultaneously replaced by their reciprocals at the same time as Z1 and Z2 are so replaced, new units are, obtained in cases III and IV which comprise reactances X1, and X2 which are the inverses respectively of those obtained in cases I and II.

On the other hand, each of cases I to IV is itself subdivided into two kinds accordingas to whether the two cut-off frequencies which limit the pass band are, as to one' of them, a critical frequency of Z1, and as to the other, a critical frequency Z2 (Z1 and Z being then of the same degree) (case A), or as to whether they are both critical frequencies of; the same reactance, for instance Z1 (Z2 being then of a lower degreethan Z2): (case B). I

A critical frequency of a reactance X is a frequency for which X=0 or X:

The degree of a reactance is the degree of the rational fraction which represents this 'reactance, that is, the highest exponent of fre quency appearing in the numerator or denominator of the fraction expressing that reactance.

To eachof cases IA to IVA, as above defined, there corresponds a unit according to the invention, comprising in relation to the equivalent ordinary unit of the type above mentioned, a reduction of one unit of the number of; resonant circuits, in one of the branches which is transformed while. to each of" the cases 113 to IVB there corresponds a unit, according to the present invention, comprising inrelation to the equivalent unit of the mentioned ordinary type, a reduction of two resonant circuits, such, reduction affecting either two circuits of the same branch or a circuit in each of the twobranches. Case IA.In this case the reactances ofv the mentioned usual, type of unit are expressed; as follows:

Z aw-+1112). (p -we); (r l-vi) (p iiser) where A1 and A3; are positive constants, and

on, (be w2m+2 represent the angular frequen:

cies arranged in the. order of increasing values.

'Reactance Z1 may therefore be realised in practice in the ordinary. parallel? typeoil-circuit diagram, considered by way of example; by

setting in parallel an inductance, a capacity. and n resonant circuits, and the arrangement formed.

by a capacity in parallel with p resonant. circuits, may, if the capacity is notftoo small, be.

Fig. 4 illustrates a filter according to my invention The unit, according to the present invention, which .is equivalent to a unit of this-type IA, is obtained by substituting for one of the two oppositely positioned sets of equal reactances, Z for instance, two unequal reactances X1 and Xz which are determined, as will be explained below, in such a Way that one of them shall comprise, when constructed of parallel resonant eircuits, oneresonant circuit less than the reactances of the setof reactances replaced thereby.

Figure 5 illustrates a non-limitative example ofan embodiment of a unit of this type of case I-A.

Case IB..-The reactances Z1 have the same expression as, in the casev IA, the reactances Z2 being of the value equivalent to the ordinary unit of this type, is

then formed by replacing the branches of the highest degree, namely Z1, :by twou-nequal reactances X1, and X2, the constituent circuit ele-.

ments. whereof are determined as hereafter described.

Figures 6 and/7 represent, by wayof nonlimitative examples, modes, of construction of the units of type IBaccording to the present invention,,fur nished with resonant. circuits, the gain. of: number of resonant circuits, as comparedwith the comparison unit of; the ordinary type, being one circuit: at X1 and one circuit at m for Figurev 6; and two circuits at,X1 for Figure 7;

Case II-A.,In thiscase the reactances of the comparison type of-ordinary unit with two pairs of equal reactances are-expressedas follows:

previous case, andthe. "reactance Z is. equal to:

t has neresonantv circuit less than Z1 -andv it can be constructed, either by n1 resonant circuits in parallel and one inductance. (Figure 101,01: by putting inrseries. with an inductance of a group. of rt -2165011811111 circuits, in parallel, a capacity andan inductance. (Fig. 11

The corresponding, ordinary. types of, filters.

may be represented by Figures 14 and 15'.

Equivalent units, according to the present in vention are ,producedby. replacing thetwo equal is employed such an ordinary type of unit as heretofore used, furnished with, resonant circuits.

reactances Z1 of the, ordinary types of units by two reactances X1 and X2 determined as here-, inafter explained, whereby it "is possible to gain, as compared with an ordinary type of unit, either the elementsoftworesonantcircuits at one of said reactances, or. in more generalized manner,

the: elements of :one resonant circuit-1 atpeach of them. i I .1;

It is to be noted that the transformations of the reactances which comprise quartz piezoelectric elements cannot'be realised except when the capacities in parallel. infthecircuitoi X1 and X2 do not fall belowa certain value.

It is clear that the circuit diagrams shown, which correspond to onoi the possible modes oi realisation of the'reactances, do-notlimit the applications of the present invention; -because X1 and X1 may be obtained, according to the known rules of the synthesis of dipoles, in the form of a graduated structure in series with a transformer; which structures make it also pos sible, according to the present inventiom'toreduce the number of constituent elements of the units correlatlvelywith the reduction of degree of X1 and X2, with reference to the, degree; of the reactances Z1 of the ordinary type of unit which they replace. T

For instance, in case II, the reactances Z and 22 may consist of theprimary-, of a transfbrrherg. the secondary of which is. closed by, a group in parallel of n-1 or n2 resonanttcir-r. cuits, and of a capacity. (either of 1 z -1 or n-2 quart; piezoelectric elements and possibly a condenser); the-"present invention then makes it possible to obtain relatively to Z1, ..in the case He, the gain oi-a resonant circuit or a quartz element art/X1 and X2, and in case 'IIb, again of one circuit at X and X2 or of two circuits at X101'X2. U h

However, in the case of the circuits which comprise-"transformers, the ordinary type of. unit makes it possible to use :common secondary circuits, so that under these conditions, the new filters become less important.

We shall now describe the method whichmakes it" possible to produce filters according, toqthe present invention, and indicate the mode for determina'tionof theicircuit elements of such filters incasesI-Aand I-Bp '0 ;-iCase.-I-A.If it is desired to replace the arm Z :by X1 and X2, it is'ynecessary to; solve the Equation 1 by means of reactances. 1 :iLet us take:.

(6) WWW flows) ue+ i w+wow ins-1(1 0 has n? T Y". i K i) whichv are distributed, according to the theorem of the reactance appliedtdh-i-Zz, in accordance with the law:

' as they have zeros The theorem of the reactance shows that:

be written in the form which satisfies Equation 1 "formally, Tn+1(p designating a v polynomial of (vH-l)" degree in most at 1: whose coeiTlcient of the termof. the highest degree is supposed equal to unity, and It being a positive constant. The residue" of a function of f in the-vicinity of Z=Zo,- is defined by stating that if a function ,f( represented by a power-series in has a pole (becomes infinite) for Z;Zo, then the coemcientof th'flrst power-term .5. 1 r t-1-- in the immediate vicinity of 2:20 is called its "residue forthis pole. The residue can also" be defined in terms of the integral of -the function around a small circle surrounding the pole. For all the poles of Z2, X1-1- Z2 and X2 Z2 must have residues at least equal to those of Z2,

or of Z1 and Z2. It. follows from'l) that they must-be equal; hence, Tn+1 111 will become zero for all the poles of Z2, so thatr :.-1 1 L; r a.

. i *PWMHWU I (n) ('p*+w:,+ .mup=+-w.rowa ){a Xvi-Z5? Flynn-(1??) r; r (P+w3-+:) 5[(P+ i) s+ ('i ?+w .+z) U-l If X1 Z: and X: Z: are reactaricesiasthey have all the poles of Z: with equal residues, and Z1 has only zeros for zero and infinite frequencies,- the same-applies to X1 andgl a It is therefore sufilcient to express X1+Z2 and Xa+Zzas*reactances. 1 1

Inasmuch as, for le -=0, they are identical vwith Z1+Zz, they cannot. lose the character of reactance before, due cto the increase 6! 46?, dzher appears fortheirr denominator, set equalertor 0', arootreither infinite or zero, or common with-a root of W2n+1 (11*):0. But according to (73), for k? 0, the'roots of h i .(p+wl)V, iz +jeii nwn iq are .separated -.-i'rom one another .and from in.- flnity by the roots of Winn (21') =0. i 1

Consequently, if k is made to increaseistarting from zero, it will :not be posible to meet any infinite root. It is-' therefore thezero root alone which has to be considered; it will be obtained for the valuelo of is 'given' by: Y

genk increases from to ii :1 increases from e1 to in-+2- u -The smallestvalue'fiof k which alone has to be retained, therefore corresponds to the smallestof the values of 0' namely, 0'1 and calling this value k1, we have:

' which is a reactance: {Cpl-is an increasing functioni 015' p? when an) :increases'efrom an to enlnkewi e it at ras r-(p) given, e1" depends from 'deereasinurommi to ed-when in Order' -that'm :shaliIbe-less thaniinf that decreases t-rcm to 0. --it.-is -therefoe:necesaary,

shall be less than the-value vHi: corgespondinarto mlcpa =0 as expected. L

. The reactances ZrrandcZa are q formedf asz-v we have seen, by placingsin parallel anginductancegza capacity and-n resonant circuits,:e aninductance and n quartz piezoelectric elemeri .and='by:rnak.= ingk =k1 we obtain a resonan eircuittzat'iixa which realised.sby;rp ttin .mg pgrallele an; in"; ductance, a capacityand n-:.l. restinantricircuits. Or an inductance and n--1 quartatpiezoelectric elements. v X1 remains of'the sametype as Zf'i. If I,

A, there appears simultaneousiy ai pole of Xrtot p=0; this eliminates the: paralleliinductance apnearing in its circuit diagram.

gi l];

the only possible reduction "is the.lei hninationlct the parallel inductance of' xi for'k =ko.

In iact ithe condition k1= ko will generallybe (complied-within narrow band filters.

" In'eifect. k0, is greater than Itis'ther'efore:siiflicieniithatz equations;

On the other hand, if P is the propagation constant, we obtain by calling P9 its value for the infinite frequency I a: *L

I A1 and consequently; I I I I I fim-m Therefore, we can suppose P 3 nep ers (1; neper=8.686 decibels) therefore:

g n- 4w or 0.s

- In order todetermine the values of X1 and the following equations are used:

l (P 1-' (P +2) n I 1 1U 11.- 2 n) (p 'i fi n+ (p n+2) n by substituting therein for k the value of 101 if we call pNz'(p and D201) the numerator and the denominator of X2, we can put down L1L-1 and Mn being respectively pin-1 and 12 degrees in 11 The determination of the elements of X1 and X2 is obtained by resolving the admittances is 7 X1 X: into their; simple elements, according to the classical method, the zeros of X1 and X2 having :been determined graphicallyv or by some s it bl method of approximation.

If it is desired to efiect the substitution at the branches Zzinstead of at the branches Z1, we have tosolve, by means of reactances S1 and S2,

1 1 (24) 1+ 1 2+ l l+ which case can be treated like the preceding one.

(26) A1 was-w:

which becomes acceptable if a 27 kznil 1 which is a condition generally realised, and the application of which in the formulae (25) makes it possible to reduce by two units the degree of S2 (representing a gain of one resonant circuit as compared with X1, and the degree of S1 which remains equal to that'o'f X1).

Let us now consider case IB, namely the unit expressed by: I

ZI 12 (2 i) (2 n) (p=+ i)(p (p+ i+r)1 that is,.the reactance theorem applied to Z1+Zz. For the equivalent unit (X1, X2, Z2) we have to Y which equations satisy Equation 1 in form and indicate that X1+Z2 and X2+Z2 admit all the poles (infinite values) of Z2 with residues equal to those of Z2: and E are parameters which are chosen as will be shown later.

"cameos 11 i In order that X1+Z2 and X2+Z2 which, e, g being arbitrarily fixed, are identical with Z1+Z2 for k =0, should remain :reactances when k increases, it"is necessary and sufficient that the ice we eliminate: k (p +e then we find w2n(p =0,

as was to be expected.

If now we eliminate Un and Vn-l we find:

U 1(I E+ i)(p %n+1)+ 2(P E 55 being given, it is necessary to choose k so that G(p shall admit a root r of Wznqfi) and it is the smallest value oi k corresponding to difierent roots c1 c2 o' 2n+1 which alone concerns us.

If k is made to increase from 0, the roots of G(p ):0, take values from w1 and w zn+1. Inasmuch as G-(w1 and G(-w 2n+1) are positive, it follows that -w1 and w' 2n+l are external to the interval limited by the modified roots; these roots therefore tend towards each other while the root which was equal to w.1 approaches u1 and the other root approaches -0' l2n T-hesemodified rootsare the roots of Glp (Equation 38) in which the parameter k appears when the value of k is increased from zero.

The roots will always continue to approach each other if we cause k to increase, because the root can correspond to onlyone value of k The roots cannot cease to be real before they are identical, so that the smallest value of k which causes a reduction of degree, will be obtained by replacing p in (38) by the values: 0'l or (-0' 2n') from which we get the characteristic 7 a1 is near m2 and O'Zn to can and that they respecdecreases. 61 and 6 211 therefore, decrease simul- 12 taneously (w x is supposed nearer (01 than M211 and nearer hys-i than wi 1 6 2n is equivalent to l in-H12 The smallest of the values k ll 'andk 2nfO1 their common values will render possible an effective reduction (a gain of one resonant circuit) if it does not exceed the value:

We have now to discuss separately the two cases ale= +l and (A) We must distinguish according to the position of 5 If E T Zn-EQLIBIlTiODS 39 become:

hence:

( ea: E 2

Now-it is easy to see'that' the value I61 complies generally with the conditions,

kl lCO ,lC1 (k -:'1)

The same evidently applies to k zn if-it can be used, that is to say, if less than 701 kl is in factless than Now,

reaches-e) If we suppose a fairly regular distribution of the characteristic values, wz -wi is of the order of w 2n+l-w2 and 6 of the order of a 2 an-H if p 3 nepers (1 neper=8.686 decibels), which is admissible, we have en ra ly 3 Y the condition here is f assists and o! the s eer 01 Likewise: 5.:

is less than 1 5 j \/m and conseq 'fsfey less than X. w Q w 2 2.5? 4 which is greater than -12, ii the pass band is not excessively wide.

. We theref re see that by taking the smallest of the values 1e1 km. we obtain a reduction of one resonantjeircuit and that, if these values can be made equ we obtain a reduction of two circuits on thelaggregate of X1 and X2. 1

Let us the elore compare k1 and h m. k1 is less than was if:

there is a min a given by (46) for which k iii i-and'leha become equal, their com- 1- enue 2 12 7 f 35 Ash Xi and 70 2s, we gain a resonant circuit. at X1 is well as Xz, subjeet to the condi- 0 lr albhjtzn is evidently always less than R12; kfiisequaltokznfor: Y

where 1 must be realised.

We can see as beforethat, if n22, we obtain;

We therioifegain, in this case; one resoiiaiitf circuit at X1 well as-at X2.

While thi; gein in the preceding case assurne ci the condition 1 l at 7o l Therefofg the condition 1 i i iii X :13 The reduetion concerns ii -(k1?) and Xmas) i jHere, kn is of the ordetjof and will generally be less ithan l la es.

Likewise, fm km will generally .be less than X it :n 3 7 If these conditions are lled, there is for the values 5m, km a gain 0 two resonant circuits i n F et-wa h l A a kl .JiWi fn w g g- Reduction of one resongn't circuit at Xe.

Rtekiuetin of one i'esonant eircuit at' 5- 5-41, 'con'smeratidn of 1 these cases is easier. r 'We have 1-? I01: and general conditions hereinbefore'indio'atedv be less than 1.

a ty mat:

hence t .15 km and 13 are positiveii 9 29a Upon comparing the results of the various cases considered, 'wezconclude that whatever the value of relatively to 1 and to 2,. a gain of two resonantcircuits is always possible under the very general conditions which have been assumed, such gain being one circuit in the unit X1 and one circuit inthe unitXg. .The gain of two circuits in the same reactancei is realisable under more restrictive conditions. It is to be observed that if the conditions are not fulfilled, the realisation for a value to be determined of :5 of the equality of the two smaller values of the value k1, k 21i, 1,

-.a.lsq all w si t on in hei m 6 "fu ments, but a less substantial one;

It will easily be seen that it is possible to effect the transformation at the reactance Z2 of the smaller degree.

What is claimed is: 4

1. An electrically symmetrical band pass filter characterized by the fact that it consists of a lattice cell comprising two opposite identical branches and two/opposite unlike branches, and that it contains a total of "four inductances and our capacitances and y mu m e resonant circuits, m being the total number of zeros and poles of the characteristic attenuation function inside the pass band, whose characteric mpedance Z varies; as a f on of ea eular frequency accordinei e fqnnu a- Z= -71 N (PT aXfl-iwis "if b designating the quantity it: and .0 being a com .6 stant. and whose attenuation function is 4 tanh where H is a constant aria vn=(p=+wa(p +to i (was!) (02, m3, w2n+1 being the angularf-requencies between to and w; and arranged in the order of increasing values, for-which the characteristic phase shift takes values which are multiples 0! Jr, said filter being characterized by the fact that the equal and opposite branches are made up of reactances equal to liflgfleiU and that the other opposite branches are. re.- a ieneesdeflee br:

x (H U,.+k?V,

" ta etmvt seamen m P('HUnkiV,.

PMWHtween)' being a censtam'eq al 0 7 and 0'1 being the angular frequency between w.

and w, for which the following expression becomes zero:

the reactance :cz including .then, under the rollowing condition, y

3 2 2 g @a e, 2n+l b36022 m resonant circuits, m being the total number of zeros and poles of the characteristic attenuation function inside the pass band, having a characteristic impedance Z which varies as a function of the angular frequency w according to. the formula:

p designating the quantity ii, and C being constant, and an attenuation unction the real part of P being equal to the characteristic attenuation and its imaginary part or chareeter st P a ii hich ar e a u the 9 tor ise to th i mula to an inductanca as well as those of the equal 17 between o. and-w and arranged in the order of increasing values, for which the characteristic phase shift assumes values which are multi-' ples of 1r, said filter being characterized by the fact that the opposite branches; equal, are made up of reactances equal to t H and that the other opposite branches are react-' ances definedby:

H( ;(2 1+ tout anteater n) K in-1'1 being a constant equal to 02n+1 being the angular frequency between wt and w; for which the following expression becomes zero:

Hup +wr vr+(mauve the reactance S2 then comprising, under the following condition:

necessary for the filter to be physically realisable, (n1) resonant circuits only, in its "shunt realisation, in addition to an inductance and a capacitance, while the other branches comprise 11 resonant circuits in addition to the same ele ments.

3. An electrically symmetrical band pass filter characterized by the fact that it consists of a lattice cell comprising two opposite identical branches and two opposite unlike branches, and that it contains a total of four inductances and four capacitances and 2m: 1sin 2 resonantcircuits, m being the total number of zeros and poles of the characteristic, attenuation function inside the pass band, having asa characteristic impedance function i I z= 2 '2 I 2 2 we at a) and as a characteristic attenuation function:

P 'HU 1 tanh 2 characterized by the fact that the equal opposite branches are made up of reactances equal toand that the other opposite branches are, react-1 saidreactan ces eachcomprising (n-1) resonant circuits, in their shunt realisation, in addition branches, the parameters 1cm, (m and the choice of the sign of c beingdetermined as follows: Let -11 and -0'2n be the zeros of:

located respectively between w; and -w2 -w2n and ll1'fl and 61 6211 defined by the following conditions must be fulfilled for the filter branches to be physically realisable:

4. An electricallysymmetrical band pass filter characterized by the factithat; it consists of a lattice cell comprising two opposite identical branches and two opposite unlike branches, and

that it contains a total of four inductances and four capacitances and mn 2m- 1 S1112? resonant circuits, m being the total number of' the equal opposite branches are made up of reactances equal to CH U,, I and that the other opposite branches are reactances x1, x2 defined by: i

x 1 H Un+km p +eaJvH F CH(p rer)(r asons-rumsaw ag r-nae located respectively between -wa and w2'*, -w2n and 3 and 61 5211 being defined by:

e l ram 3" \/(@;.t.-- a) the reactance :m comprising, in its shunt realisation, n2 resonant circuits, in addition to an inductance, while the :01 reactance comprises n such circuits, the following conditions having to be met for the filter branches to be realisable:

5. An electrically symmetrical band pass filter characterized by the fact that it consists of a lattice cell comprising two opposite identical branches and two opposite unlike branches, and that it contains a total of four inductances and four capacitances and we: wi

will): wi

7 finial-sin resonant circuits, m being the total number of zeros and poles of the characteristic attenuation function inside the pass band, having as a characteristic impedance function Q p r e) (i -W 5 C being a constant, and as a characteristic attenuation function tanh -H p2+wa2 U11 and V11 being defined by the equations Un=(p i)(p i) (p+ in) said filter being characterized by the fact that the equal opposite branches consist of reactances equal to:

c p=+we m p where H is a constant and in that the other opposite branches are reactances defined by:

ki being defined as in claim 2 and having to meet the same condition as in said claim, th reactance :m then comprising in its series realisation, an inductance in series with the parallel grouping of a capacitance and resonant circuits, (n-1) resonant circuits only whereas in the circuit diagram of the same type, the other reactances comprisen such circuits.

6. An electrically symmetrical band pass filter characterized by the fact that it consists of a lattice cell comprising two opposite identical branchesand two opposite unlike branches,-,and;-

that it contains a total of. four inductances and four capacitances and.

' I mn 2m-- 1 sin resonant circuits, m being the total number of zeros and poles of the characteristic attenuation function inside thepass band, having as a characteristic impedance-function:

\/(P a )(1 a) :p I C being-a constant,fand. asia characteristic attenuation function:

k 2n+1 being a constant equal to ll 1 m H Gavel 41 where H is a constant and a2n+i is the angular frequency between. wt and we for which the value zero, is attained by the expression vfl= p +wz p +w2 (Meal) and having to meet the same condition as in said claim, the reactance S2 then comprising, in its realisation by series connection of an inductance with the parallel grouping of a capacitance and resonant circuits, (n-1) resonant circuits only whereas" in the circuit. diagram of the same type, the other reactances comprise n such circuits.

7. An electrically symmetrical band pass filter characterized by the fact that it consists of a lattice cell comprising two opposite identical branches and two opposite unlike branches, and that it contains a total of four inductances and four capacitances and resonant circuits, m being the total number of zeros and poles of the characteristic attenuation function inside the pass band, having as a characteristic impedance function:

arm

and as a characteristic attenuation function:

'tanh w; 2 visa/wander r) as, am and the sign of 2 being determined as indicated in said claim, km and e m having to meet 7 resonant circuits, m being the total number of zeros and poles of the characteristic attenuation function inside the pass band; having as a characteristic impedance function and as a characteristic attenuation function:

U11 and Va being defined by the equations =(P+ )(P-l- 3) (P -t in) V v..= p*+-: (p +w:) (when) said filter being characterized by the fact that 22 the equal and opposite branches consist ot reactances equal to H m-new aWH 7am and 111 being defined by the equations a=+a' 1 2n dn E, it- 5,. -01 and- 47211. being the zeros of H (2 (P -t m) i-r located respectively between w.,, and --w, w,, and -w; and 62 and 63,,

being defined by v eat/F rm ;n 6 in) (dflf' u and having to meet the sameconditions as in said claim, the reactance x: then comprising in its realisation by connection in series of an inductace with the parallel grouping of a capacitance and resonant circuits n-2 resonant circuits only while an comprises n such circuits, in arealisation by a diagram or the same type.

ROBERT PIERRE LEROY.

REFERENCES CITED The following references are of record in' the file of this patent:

UNITED STATES PATENTS Number Name i Date 1,996,504 Darlington Apr. 2, 1935 2,037,171 Lane Apr. 14, 1936 2,115,138 Darlington Apr. 26, 1938 2,222,417 Mason Nov. 19, 1940 

